The short summary: In a chase the last initiative slot gets to assign advantage or triumph that effects others negatively on the next competitive check, but other slots can only boost themselves. Especially potent since the last slot can spend a triumph to upgrade the next competitive check.

First my understanding of how chase works:
1. Pre-chase: Initiative roll (vigilance), assign slots.
2. Chase: Competitive check.
2a. Spend advantages/threats from competitive check to add boost/setback to next “check”.
3. Initiative order “structured play”.
3a. Spend advantages/threats from action to add boost/setback to next “check”.

*In the above: 2a. The next check is the action taken during initiative order. But for 3a. the “next check” for the last initiative slot is the upcoming competitive check.

In a quick foot-chase example: (for simplicity 1 PC, 1 NPC)
1. Initiative roll (vigilance), PC first slot, NPC second slot.
2. Competitive Athletics check, PC wins with 2 success and 2 advantage. NPC has 1 success with 2 advantage. PC wins, and is escaping so moves away one rand band.
2a. PC decides to spend 2 advantage to give setback die to NPC’s next roll.
2b. NPC decides to spend 2 advantage to give setback die to PC’s next roll.

3. Initiative order:
PC: Takes a maneuver (stimpack) and then and action (attack) with 1 setback from above. They succeed and have another 2 advantage.
3a. They spend the advantage to give a setback die to the NPC’s next action.

NPC: Takes a maneuver (draw weapon) and then and action (attack) with 2 setbacks, one from the competitive roll, and one from the PC’s attack from above. Lets say they fail but have 2 advantage.
3b. They spend that advantage to give a setback or advantage die to the next check(roll), which happens to be the competitive Chase check.

What I am seeing is this: This means that the Last slot(NPC above) has a chance to add positive or negative dice towards the next competitive check (which in a chase is arguably the most important roll) while the first slot(PC) can only add a boost. Some might think no big deal its only a boost or setback, but if the last slot earned a triumph they could also use it to upgrade the opponents next check, aka the competitive check.

My questions:
1. First and foremost, am I calculating the chase checks and the bonuses/setbacks correctly?
2. Is this imbalanced? Especially with triumph upgrades.

1. Looks correct to me.

2. Personally I've never had a problem with the chase rules. There are talents that remove setback dice from both Athletics and Piloting checks, so they really aren't that big of a deal. Also, Triumphs and Despairs don't happen all that often, so it's hardly game breaking. But sure, you could see it as the person acting last in the regular initiative sequence would have an advantage in the chase portion. However, that is balanced by having the other party act first in the regular initiative order, where a well-aimed shot could put an end to the whole thing.

I think in this case, you're being to literal with the "next check." You never just pass an upgrade to the next guy, and you shouldn't just be passing boost dice either (though there's a lot more of them, so I can understand sometimes letting the creative spirit rest a little).

"Okay, you rolled a Triumph, what would you like to do with it?"

"I'm going to upgrade my competitive check!"

That's lame.

"Triumph! What happens?"

"I grab a trash can lid, toss it at the top of the stairs and then slide down like a badass!"

"I grab a trash can lid and throw it at the lead Stormtrooper like Captain America, sending him staggering!"

"I grab a trash can lid and realize the trash can is a secret entrance to Fraggle Rock! I say, 'Guys, quick! In here!' We jump in and close the lid!"

In the first case, he's going to upgrade his own check. In the second, he'll upgrade the difficulty of his opponent's check. In the third, he's taking some narrative control, which is often the coolest thing to do.